Nonstandard approach to gravity for the dark sector of the Universe

0.91 MB·English

by  P. C. Stichel

Nonstandard approach to gravity for the dark sector of the
Universe∗
P.C. Stichel1) and W.J. Zakrzewski2)
1)An der Krebskuhle 21, D-33619 Bielefeld, Germany
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e-mail:[email protected]
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0
2
n 2)Department of Mathematical Sciences, University of Durham,
a
J Durham DH1 3LE, UK

8 e-mail: [email protected]
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]
c
q
Abstract
-
r
g We summarize the present state of research on the darkon fluid as a model for the dark
[
sectoroftheUniverse. Nonrelativisticmasslessparticlesareintroducedasarealizationofthe

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Galilei group in an enlarged phase space. The additional degrees of freedom allow for a non-
v
6 standard,minimalcouplingtogravityrespectingEinstein’sequivalenceprinciple. Extendedto
8 a self-gravitating fluid the Poisson equation for the gravitational potential contains a dynam-
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4 ically generated effective gravitational mass density of either sign. The equations of motion
. (EOMs) contain no free parameters and are invariant w.r.t. Milne gauge transformations.
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0 Fixingthegaugeeliminatestheunphysicaldegreesoffreedom. TheresultingLagrangianpos-
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sessesnofreeparticlelimit. Theparticlesitdescribes,darkons,existonlyasfluidparticlesof
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: aself-gravitatingfluid. Thisdarkonfluidrealizesthe zero-mass Galilean algebraextendedby
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i dilationswithdynamicalexponentz= 5. WereducetheEOMstoFriedmann-likeequations,
X 3
derive conserved quantities and a unique Hamiltonian dynamics by implementing dilation
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a symmetry. BytheCasimirofthePoisson-bracket(PB)-algebrawefoliatethephasespaceand
construct a Lagrangian in reduced phase space. We solve the Friedmann-like equations with
the transition redshift and the value of the Casimir as integration constants. We obtain a
deceleration phase for the early Universe and an acceleration phase for the late Universe in
agreement with observations. Steady state equations in the spherically symmetric case may
model a galactic halo. Numerical solutions of a nonlinear differential equation for the grav-
itational potential lead to predictions for the dark matter (DM) part of the rotation curves
(RCs)ofgalaxiesinqualitativeagreementwithobservationaldata. Wealsopresentageneral
covariant generalization of the model.
∗Acceptedforpublicationby‘Entropy’inthespecialissue‘Modifiedgravity: Fromblackholesentropytocurrent
cosmology’
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Contents
• 1 Introduction 3
• 2 Galilean massless particles and their nonstandard coupling to gravity 5
– 2.1 Galilean massless particles 5
– 2.2 Coupling to gravity 8
• 3 A self-gravitating (darkon) fluid 9
– 3.1 Lagrangian formulation 9
– 3.2 Relabeling symmetry and the transport equation for the gravitational field 11
– 3.3 Eulerian formulation 11
– 3.4 Coupling with baryonic matter 12
• 4 Hamiltonian dynamics 13
• 5 Space-time symmetries 15
– 5.1 Space-time translations and the energy-momentum tensor 16
– 5.2 Zero mass Galilean symmetry 16
– 5.3 Angular momentum 17
– 5.4 Anisotropic scaling 17
• 6 Darkon fluid cosmology 18
– 6.1 Cosmological EOMs 18
– 6.2 Hamiltonian dynamics 19
– 6.3 Solution of the EOMs 24
– 6.4 Distances 26
– 6.5 Predictions versus observations 27
– 6.6 Comparison with alternative models for dark energy 29
• 7 Modeling dark matter halos by a steady state darkon fluid 31
– 7.1 Equation for the gravitational potential 31
– 7.2 Predictions versus observations 34
• 8 Influence of the cosmic expansion on binary systems 38
• 9 Outlook for a general covariant theory 40
• 10 Conclusions and outlook 41
2
1 Introduction
Understanding the dark sector of the Universe is one of the greatest challenges of today’s
theoretical physics. The main question is whether the underlying astrophysical observations
(missing gravitation in galactic systems resp. the accelerated expansion of the Universe)
can be understood by means of known physical concepts, e.g. General Relativity (GR), or
whether some kind of “new” physics is needed. Indeed, cosmic acceleration could be an
apparent effect due to the averaging over large scale inhomogeneities in the Universe, but a
reliable quantification of such an effect is not yet available (see the recent review [1]). On
theotherhandthereexistsanoverwhelmingevidencefortheexistenceofgravitationaleffects
on all cosmological scales (termed “dark matter” (DM)) which cannot be explained by the
gravitation of standard matter in the framework of GR (see the review [2]).
ThecurrentpictureofcosmologicalstructureformationassumesforDMsomepressureless
dust containing massive, perhaps weakly interacting cold particles (CDM). The present stan-
dard cosmological model (ΛCDM-model) uses besides CDM a positive cosmological constant
ΛasthecausefortheacceleratedexpansionoftheUniverse(seerecentreviewsondarkenergy
(DE),e.g. [3],[4]and[5]). But,“favouredbyanumberofobservations”[3],theΛCDM-model
suffers at least from the following insufficiencies:
• Interpreted as the energy density of the vacuum, the experimental value of Λ turns out
to be a factor of about 1054 too small (see the recent estimate in [6]).
• None of the proposed CDM-constituents has been observed (cp. [7]).
• Rotation curves (RCs) of DM-dominated galaxies behave in their inner part in sharp
contrast to the CDM-based simulations (known as the core-cusp problem [8]).
In a very recent paper P. Kroupa [9] lists a large set of extragalactic observations which
falsify the ΛCDM-model.
In this paper we will neither comment on the huge number of phenomenological models
foradynamicalDE(see[3]-[5]),whichallrestonatleastoneunknownfunction(e.g. ascalar
field potential), nor on those modified gravity theories which explain DM-effects only (e.g.
modified Newtonian dynamics (MOND) in its simplest, Milgromian form [10]). But we know
of at least two kinds of modified gravity models which explain both, DE and DM i.e. the
whole dark sector of the Universe:
• ForMOND-modelsseesection9oftheveryrecentreview[11]. Butallthesemodelsare
very phenomenological in that they are based on one unknown “MOND function”.
• Modified theory of gravity (MOG) by Moffat [12] in which, besides the Einstein-Hilbert
andmatteractions,amassivevectorfieldisintroducedwhosemass,couplingconstantto
matterandgravitationalconstantarepromotedtoscalarfields. Theself-interactionpo-
tentialsofthesefourfieldsarenotfixedaprioriandplaytheroleofDE[13]. Thismodel
can explain galactic phenomena as e.g. RCs [14] as well as the accelerated expansion of
the Universe and other cosmological observations [13].
In summary, all present attempts to explain quantitatively the new cosmological obser-
vations (e.g. late time cosmic acceleration) as well as the galactic RCs and other galactic
phenomena contain either some new parameters or even free functions. If we assume that
the cosmic acceleration is a real effect and not an apparent one (see above) then, obviously,
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we need some new (i.e. unconventional) physics which, however, should be based on known
fundamental physical principles (e.g. symmetry).
In this paper we review some ongoing research on a model which is a first building block
foranewtheorydescribingthedarksectoroftheUniverse. Thismodel,introducedin[15]and
furtherdevelopedin[16]and[17],containsnonewparametersinitsLagrangianandisbased
on well- known physical principles (Galilei symmetry and Einstein’s equivalence principle).
Theonlyfreeparametersappearinginourmodelaresomeintegrationconstants(seesections
6 and 7). The basic idea it uses involves nonrelativistic massless particles. The use of such
particles may appear very strange at a first sight as it seems to contradict Special Relativity
according to which massless particles must move with the velocity of light. However, as
we have shown before, we can introduce nonrelativistic massless particles if we enlarge the
dimension of phase space; i.e. such particles are not described by position and momentum
only. The additional degrees of freedom allow for a nonstandard, minimal coupling of these
“exotic” particles [15] to gravity in accordance with Einstein’s equivalence principle. This
coupling, for a self-gravitating fluid, leads to a dynamically generated effective gravitational
mass density of either sign which is the source of the corresponding gravitational field. This
fact leads to the possibility of using such a model as a first building block of a new theory
describing the dark sector. In our approach we formulate the equivalence principle (local
equivalence between gravitation and acceleration) as usual as an invariance of the dynamics
w.r.t. arbitrary time-dependent translations (Milne gauge transformations). By fixing the
gauge we can reduce the dimension of phase space and remove the unphysical variables. The
Lagrangianoftheresultantmodeldoesnotpossessafreeparticlelimitandhencetheparticles
itdescribes,calleddarkons,existonlyasfluidparticlesofaself-gravitatingfluid. Thisdarkon
fluidpresentsadynamicalrealizationofthezero-massGalileanalgebraextendedbyanisotropic
dilations with the dynamical exponent z = 5. This dilation symmetry emerges in a natural
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way from the minimal gravitational coupling. The cosmological EOMs are derived, as usual,
fromthedarkonfluidEOMsbyrestrictingtheformoftheirsolutionsthroughthecosmological
principle. However, the corresponding Hamiltonian cannot be derived from the darkon fluid
Hamiltonian. So we have to derive a Hamiltonian and the corresponding Poisson-brackets
(PBs) from the EOMs, a procedure which, as is well known, is not unique. So we construct,
using the constants of motion, a Hamiltonian which has the correct scaling dimension and
obtain the unique PBs by implementing the dilation symmetry. Moreover, our procedure
allows us to construct a Lagrangian having foliated the phase space by means of the Casimir
of the PB-aLgebra. We solve the cosmological EOMs and obtain, by fixing two integration
constants by their values determined from experimental data, a prediction for the Hubble
parameter H(z) (z is the redshift) in general agreement with the existing data.
Furthermore, we demonstrate that our model not only gives a proper description of the
late time cosmic acceleration but it also predicts galactic halos and, qualitatively correct, the
corresponding flat rotation curves (RCs). To do this we derive for the spherically symmetric
steady state case a nonlinear ordinary differential equation for the gravitational potential
which is then solved numerically.
Almost all cosmological models are based on General Relativity (GR) or on some of its
modifications which, however, preserve general covariance. The appearance of anisotropic
dilationsinourmodelseems,atfirstsight,toforbidageneralizationofourmodeltoageneral
covarianttheory. Butthedilationsymmetrywithz= 5 isanemergentsymmetryandnotan
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input of our model. So we are able to propose a generalization of our model showing general
covariance without imposing any form of scale symmetry.
One of the aims of the present paper is to show in detail the main ideas of our model. A
comparison with observational data (Hubble parameter resp. galactic rotation curves) serves
only as an illustration that it might be meaningful to develop the present model to a proper
theory of gravitation for the dark sector of the Universe. For this reason we do not produce
anyleastsquarefitsfortheundeterminedintegrationconstantsappearinginthecosmological
context.
Thepaperisorganizedasfollows: Insection2weintroduceGalileanmasslessparticlesand
their coupling to gravity. In section 3 we describe the generalization of this particle picture
to a self-gravitating fluid, fix the gauge and so obtain the darkon fluid. Then, instead of
the Poisson equation, we get a transport equation for the gravitational field. In section 4
we describe the Hamiltonian dynamics of the darkon fluid to have a basis for quantization.
Space-time symmetries of our model are discussed in section 5. The darkon fluid cosmology
is presented in section 6. Starting with the cosmological principle we derive Friedmann-like
equations,discussthecorrespondingHamiltoniandynamicsandcompareourpredictionswith
theexistingdatafortheHubblediagram. Insection7weconsidersteadystateequationsinthe
sphericallysymmetriccaseandderiveanonlinear,ordinarydifferentialequation(ODE)forthe
gravitationalpotential. NumericalsolutionsofthisODEleadtopredictionsforthebehaviour
of the DM-part of galactic RCs. In section 8 we outline how to determine a correction to
Newton’s gravitational law due to the cosmic expansion. Finally, in section 9 we present a
general covariant generalisation of our model. We close with some conclusions and give an
outlook for future work.
2 Galilean massless particles and their nonstandard
coupling to gravity.
Asstatedearliertheintroductionofnonrelativisticmasslessparticlesmayappearverystrange
at first sight. It seems to contradict Special Relativity according to which massless particles
mustmovewiththevelocityoflight. ButinSpecialRelativityspinlessparticlesaredescribed
by the vectors of position and momentum only. In our recent papers [15]-[17] we have shown
thatwecanintroducenonrelativisticmasslessparticlesifweenlargethedimensionofthephase
space from six to twelve. In subsection 2.1 we will show that this enlargement of phase space
is also necessary: The 12-dimensional phase space is the minimal one allowing the dynamical
realization of nonrelativistic massless particles.
How to couple gravity if there is no mass? Our nonrelativistic framework forbids the
couplingtoenergy,asthenthiswouldbearelativisticeffect. SowetakeEinstein(cid:48)sequivalence
principle as a starting point in subsection 2.2. Then, within the Lagrangian framework, the
gravitational field will be coupled to one of the new degrees of freedom.
2.1 Galilean massless particles
Our first task involves finding a minimal dynamical realization of the unextended (m = 0)
Galilean algebra G in three space dimensions.
0
5
G is given by the Lie-brackets
0
[A , L ] = (cid:15) A , A ∈(P ,K ,L ),
i j ijk k i i i i
[H,K ] = P , [H, P ] = [H, L ] = 0, (1)
i i i i
[P , K ] = 0 [P , P ] = 0, [K , K ] = 0,
i j i j i j
where H,P ,K and L are the generators of time-resp. space translations, boosts and rota-
i i i
tions.
A minimal dynamical realization of (1) involves a construction of the elements of G in
0
terms of a minimal number of phase space variables and without the introduction of any free
functions or parameters. Then the Lie-brackets in (1) have to be understood as Poisson-
brackets (PBs) w.r.t. the PB-algebra of phase space variables.
Let (cid:126)x(t) be a particle trajectory and p(cid:126) the corresponding conjugate momentum satisfying
the fundamental PB-relation
[x , p ] = δ (2)
i j ij
The translation generator P(cid:126) acts on(cid:126)x resp. p(cid:126) as
[x , P ] = δ resp [p , P ]=0.
i j ij i j
Therefore we get the identification
P = p . (3)
i i
Toshowthatourminimallyenlargedphasespacecontains,besides(cid:126)xandp(cid:126),twoadditional
vectors corresponding to the reduced boost (cid:126)q and the velocity (cid:126)y, we proceed in two steps:
• 1. Define a reduced boost vector (cid:126)q
q ≡K − p t. (4)
i i i
The boost K(cid:126) acts on(cid:126)x as
[x , K ] = δ t (5)
i j ij
leading by (2) and (4) to
[x , q ] = 0. (6)
i j
TakingthePBof(4)withp itiseasilyseenbytheexpressionsin(1)thatthePBofp
j i
and q vanishes
j
[p , q ] = 0. (7)
i j
Despite of (6) and (7) (cid:126)q turns out to be nontrivial: Taking the PB of (4) with H gives
q˙ = [q , H] = −p . (8)
i i i
Thus we conclude from (6)-(8) that the q are independent dynamical variables which,
i
according to (4) and (1), are commutative
[q , q ] = 0. (9)
i j
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• 2. Next we introduce the velocity vector
y ≡[x , H], (10)
i i
whichistranslationallyinvariant(takethePBof(10)withp andusetheJacobiidentity)
j
to get
[y , p ] = 0. (11)
i j
From (10), (1), (2) and (5) we get the action of the boost generator on y
i
[y , K ] = δ . (12)
i j ij
Inserting now (4) into (12) leads by (11) to
[y , q ] = δ . (13)
i j ij
We conclude from (13) and by (6), (7) and (9) that (cid:126)y cannot be a function of the other
variables(cid:126)x, p(cid:126) or (cid:126)q.)
To close the PB-algebra in phase space we choose, in accordance with the principle of a
minimal dimension,
[x , y ] = 0. (14)
i j
Note that (14) respects all Jacobi identities.
It remains to construct out of the phase space variables(cid:126)x, p(cid:126), (cid:126)q, and (cid:126)y the Hamiltonian H
and the generator of rotations L(cid:126).
Our foregoing results lead to the EOMs
x˙ = y , p˙ = 0 and q˙ = −p (15)
i i i i i
but y˙ has not not yet been fixed.
i
The EOMs (15) together with the canonical PBs
[x , p ] = δ , [y , q ] = δ (16)
i j ij i j ij
imply that the Hamiltonian is given by
H = p y + f((cid:126)q)
i i
The principle of a minimal dynamical realization of G forces us to put f = 0. Thus we see
0
that the Hamiltonian H is given by
H = p y , (17)
i i
which leads, due to (16), to the EOM for y , namely
i
y˙ = 0. (18)
i
The conserved generator of rotations L is then given by
i
L = (cid:15) (x p + y q). (19)
i ikl k l k l
The two parts of L act separately on the ((cid:126)x,p(cid:126)), resp. ((cid:126)y,(cid:126)q) parts of the phase space.
i
The Hamiltonian dynamics given by H (17) and the nonvanishing PBs (16) is equivalent
to the dynamics derived from the Lagrangian
L = p x˙ + q y˙ − p y . (20)
0 i i i i i i
Finally we conclude
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• AminimaldynamicalrealizationoftheunextendedGalileialgebrainthreedimensional
configuration space requires a 12-dimensional phase space spanned by ((cid:126)x, p(cid:126), (cid:126)q, (cid:126)y)
• The relation between energy H and the momenta (velocities) is nonstandard,
• Momenta p and velocities y are dynamically independent,
i i
• The q have no counterpart in standard particle mechanics,
i
• Galilean massless particles move with arbitrary finite velocity.
Dilations
G maybeenlargedbydilationswitharbitrarydynamicalexponentz. Thecorresponding
0
generator has to satisfy the following PB-relations (cp [18] for z= N)
2
1
[D, H] = −H, [D, P ] = − P , (21)
i z i
(cid:18) (cid:19)
1
[D, K ] = 1− K , [D, L ] = 0.
i z i i
It can easily be checked that D is given by
(cid:18) (cid:19)
1 1
D = tH − x p + 1− y q . (22)
z i i z i i
Below we will show that the value of z becomes fixed by the coupling of our massless
particles to gravity.
2.2 Coupling to gravity
To couple our massless particles to the gravitational field strength g ((cid:126)x,t) we have to be
i
consistent with Einstein(cid:48)s equivalence principle: Locally, the gravitational field strength is
equivalent to an accelerating frame. The only known nonrelativistic EOM for the particle
trajectory satisfying this form of the equivalence principle is given by the Newton law:
x¨ (t) = g ((cid:126)x(t),t) (23)
i i
because (23) is invariant w.r.t. arbitrary time-dependent translations (Milne gauge transfor-
mations) [19]
(cid:48)
x → x = x + a (t) (24)
i i i i
provided that g transforms to
i
(cid:48) (cid:48)
g ((cid:126)x ,t) = g ((cid:126)x,t) + a¨ (t). (25)
i i i
The EOM (23) is realized if we add to L an interaction part (minimal coupling)
0
L = −q g ((cid:126)x,t) (26)
int i i
Then the EOM q¨ =0 gets replaced by
i
q¨ = q ∂ g . (27)
i k i k
ThetotalLagrangianL=L +L nowbecomesinvariantw.r.t. thegaugetransformations
0 int
(cid:48) (cid:48) (cid:48) (cid:48)
x = x + a (t), ,y = y + a˙ (t), p = p , q = q , (28)
i i i i i i i i i i
(cid:48) (cid:48)
g ((cid:126)x ,t) = g ((cid:126)x,t) + a¨ (t).
i i i
8
This gauge invariance, resp. Einstein(cid:48)s equivalence principle still holds if we add in (26)
further terms containing spatial derivatives of g i.e. by replacing
i
g → g − κ (cid:52)g +higher order terms, (29)
i i 1 i
where (cid:52) is the laplacian and K is a constant.
1
3 A self-gravitating (darkon) fluid
3.1 Lagrangian formulation
Inthelastsubsection(2.2)wehaveintroducedamodelforthemotionofoneGalileanmassless
particle in an external gravitational field(cid:126)g((cid:126)x,t).
Now we want to generalize this model in two respects:
• 1. Instead of one particle we consider a continuum (fluid) labeled by comoving coordi-
nates ξ(cid:126)∈R3 (Lagrangian formulation of fluid dynamics, cp. [20]).
• 2. We treat the gravitational field (cid:126)g as a dynamical variable in order to get a self-
gravitating fluid.
To realize the first point we substitute in the EOMs (23) and (27)
x (t) → x (ξ(cid:126),t) and q (t) → q (ξ(cid:126),t). (30)
i i i i
To realize the second point we add to our previous Lagrangian L +L a kinetic part
0 int
L for the gravitational field. We choose as usual
field
1 (cid:90)
L = − d3x(g ((cid:126)x,t))2, (31)
field 8πG i
where G is Newton’s gravitational constant.
Then our Lagrangian, formulated in terms of (x ,x˙ ),(q ,q˙ ) and(cid:126)g, becomes
i i i i
(cid:90) (cid:16) (cid:17)
L = −n d3ξ q˙ (ξ(cid:126),t)x˙ (ξ(cid:126),t) + q (ξ(cid:126),t)g ((cid:126)x(ξ(cid:126),t),t) + L , (32)
0 i i i i field
where n is the (constant) particle density in ξ(cid:126)-space.
0
Varyingq ,resp.,x leadstothecontinuumgeneralizationofthepreviousEOMs(23)and
i i
(27)
x¨ (ξ(cid:126),t) = g ((cid:126)x(ξ(cid:126),t),t) (33)
i i
and
∂
q¨(ξ(cid:126),t) = q (ξ(cid:126),t) g ((cid:126)x(ξ(cid:126),t),t). (34)
i k ∂x k
i
Now we have two possibilities to go further:
In the first case we take g =−∂ φ and vary φ getting
i i
(cid:52)φ = 4πG∂ (nq ), (35)
k k
with
n((cid:126)x,t) = n J−1| (particle density) (36)
0 ξ(cid:126)=ξ(cid:126)((cid:126)x,t)
and
(cid:32) (cid:33)
∂x (ξ(cid:126),t)
J = det i (Jacobian), (37)
∂ξ
k
9
where ξ(cid:126)((cid:126)x,t) denotes the function inverse to(cid:126)x(ξ(cid:126),t).
TherighthandsideofthePoissonequation(35)describesadynamicallygeneratedeffective
gravitational mass density which may be of either sign with
+ sign leading to an attractive gravitation,
- sign leading to a repulsive gravitation.
This promotes the self-gravitating fluid to being a possible candidate for the dark sector
of the Universe.
Note that our EOMs are invariant w.r.t the gauge transformations (see (24), (25))
x → x + a (t), (38)
i i i
when φ transforms as
(cid:48) (cid:48)
φ((cid:126)x,t) → φ ((cid:126)x ,t) = φ((cid:126)x,t) − a¨ (t)x , (39)
i i
and q and n remain invariant.
i
In the second case we vary g instead of φ and this leads to a linear relation between q
i i
and g :
i
g ((cid:126)x,t) = −4πGn((cid:126)x,t)q (ξ(cid:126)((cid:126)x,t),t). (40)
i i
Note that the expression for g can be rewritten as
i
1 (cid:16)g (cid:17)
q (ξ(cid:126),t) = − i ((cid:126)x(ξ(cid:126),t),t). (41)
i 4πG n
Theexpression(41)allowstoeliminate,fromtheEOMs,theunphysicalphasespacevari-
ables q in favour of the gravitational field strength g .
i i
This removes the unphysical degrees of freedom. However, we have to pay a prize for this
as now
• thegravitationalfieldstrengthisnotautomaticallygivenbythegradientofapotential,
• the expression (41) destroys the gauge symmetry i.e. it fixes the gauge.
Remark The gauge symmetry can be restored by considering a curl-free velocity field
(cid:126)u = ∇(cid:126)u
and treating u as the variable to be varied [16].
Inserting q from (41) into the previous EOMs we get two coupled EOMs for (cid:126)x(ξ(cid:126),t) and
i
(cid:126)g((cid:126)x(ξ(cid:126),t),t)
d2
x (ξ(cid:126),t) = g ((cid:126)x(ξ(cid:126),t),t) (42)
dt2 i i
(initial condition(cid:126)x(ξ(cid:126),0)=ξ(cid:126)) and
d2 (cid:16)g (cid:17) 1
i = ∂ g2. (43)
dt2 n 2n i k
Furthermore by inserting (41) into the Lagrangian (32) we obtain
(cid:32) (cid:33)
n (cid:90) d g ((cid:126)x(ξ(cid:126),t),t) 1 (cid:90)
L = 0 d3ξ i x˙ (ξ(cid:126),t) + d3xg2((cid:126)x,t), (44)
4πG dt n i 8πG i
where we have used the identity
(cid:90) (cid:90)
d3ξJg2((cid:126)x(ξ(cid:126),t),t) = d3xg2((cid:126)x,t).
i i
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